Conclusions

In the present study, we constructed nuclear energy density functionals in terms of derivatives of densities up to sixth order. This constitutes the next-to-next-to-next-to-leading order (N$^3$LO) expansion of the functional, whereby, in this scheme, the contact and standard Skyrme forces provide the zero-order (LO) and second-order (NLO) expansions, respectively. The higher-order terms were built to provide tools for testing convergence properties of methods based on energy density functionals, within the spirit of effective field theories.

At N$^3$LO, depending on several options of using the energy density functionals, the numbers of free coupling constants are as follows. If one would like to include the density dependence of all the coupling constants (the option, which is not advocated here), one would have to use 658 different terms in the functional. Full functional with density-independent coupling constants contains 188 terms, while functionals restricted by Galilean and gauge symmetries contain 50 and 21 terms, respectively. If both isoscalar and isovector channels are included, all these numbers must be multiplied by a factor of two

At the present stage of searching for precise, spectroscopic-quality nuclear functionals, extensions beyond the standard Skyrme NLO form are mandatory, see the analysis in Ref. [40]. These may include richer density dependencies [41,42], higher-order derivative terms. as constructed in the present study, terms of higher powers in densities, richer forms of functional dependence beyond simple power expansions, and possibly many other modifications.

Further studies of higher-order energy density functionals requires constructing appropriate codes to solve self-consistent equations. Although this is a complicated problem, various techniques have already been developed that can be used here. First of all, expressions for mean fields must be derived by using the standard methods presented, e.g., in Refs. [25,24,43]. Obviously, such mean fields will involve derivative operators up to sixth order, so the connection with the one-body Schrödinger equation, discussed, e.g., in Ref. [28], will be lost. Nevertheless, all basis-expansions methods can still be used and their implementation will not be basically different than it was done up to now at NLO. Work along these lines is now in progress.

This work was supported in part by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme, by the Polish Ministry of Science under Contract No. N N202 328234, and by the UNEDF SciDAC Collaboration under the U.S. Department of Energy grant No. DE-FC02-07ER41457.